2.1 Epistemic Peer Disagreement

In his 2007 paper, “The Epistemology of Disagreement: The Good News,” David Christensen presents a case of peer disagreement that has become a staple of the philosophical literature on the topic.

Before exploring some answers to Christensen’s question, some preliminaries are in order. The philosophical debate assumes symmetry between individuals, so that all parties involved hold symmetric positions with respect to the body of evidence shared among them. That is, none of the individuals have extra private information the others lack that would be advantageous to their epistemic position. It’s further assumed that all individuals hold symmetric positions with respect to their cognitive and epistemic abilities. That is, the individuals involved in a disagreement are assumed to have the same level of cognitive ability and none of the individuals are worse off than any other in terms of truth-seeking capacities, possibly determined by previous track records. Satisfying these assumptions entails that the individuals party to a disagreement are epistemic peers.

Epistemic peerhood makes the problem of disagreement especially difficult since all parties would be hard-pressed to deliver a compelling reason why they should dismiss the opinions of the others. Indeed, Christensen argues that you simply cannot dismiss the opinion of another based on the fact that they disagree with you due to what he calls the independence condition: “ $\dots$ I should assess explanations for the disagreement in a way that’s independent of my reasoning on the matter under dispute  $\dots$” (Christensen, Reference Christensen2007, 199). Christensen concludes that upon meeting the independence criterion, and the fact that a peer is just as good at assessing the matter at hand as you are, you (and your peer) are rationally compelled to take the opposing opinion of your peer as evidence against your own opinion. The evidence, in turn, ought to make you less confident in your initial assessment. In the restaurant case, for instance, the opposing conclusions should make you less confident in your answer (and the same goes for your friend), where you should now give equal credence to the hypotheses (193). Revising your epistemic state in such a way amounts to conciliating, where you move your opinion toward your peer’s (and they move their opinion toward yours). In case of epistemic peerhood, you and your friend are committed to “splitting the difference” (203).

Although conciliationism, in general, does not necessarily imply that peers in a disagreement treat all the assessments equally at all times, Adam Elga (Reference Elga2007) has argued that an equal weighting of peer assessments is indeed the only rational response, which he dubbed the equal weight view. In his words,

That is, assuming that the interlocutors are epistemic peers, the equal weight view implies that the probability of each of them being closest to the truth upon learning of their disagreement should be equal. According to Elga, it is absurd to think that a disagreement supplies you with evidence that you are a better judge (486). In the absence of such evidence, you are rationally committed to thinking you and your peer are equally likely to be correct given the nature of epistemic peerhood. So, each must lend equal weight to all the assessments given.

Of course, not everyone agrees with the conciliationist attitude. In his early work on disagreement, Thomas Kelly (Reference Kelly, Hawthorne and Szabo2005) defended a stick to your guns strategy that is often called the steadfast view. We should note that there are many accounts that fall under the steadfast heading, and unfortunately, we can’t cover them all, but we’ll present two possible candidates here.Footnote ⁶ Among the most radical, the no independent weight view says “In at least some cases of peer disagreement, it can be perfectly reasonable to give no weight at all to the opinion of the other party” (Kelly, Reference Kelly, Goldman and Whitcomb2011: 186). While it’s unclear whether anyone explicitly endorses the latter view, some views might entail it. Ralph Wedgwood (Reference Wedgwood2007), for example, has argued in favor of privileging the first-person perspective such that one embraces an egocentric bias, or having a fundamental trust in one’s own cognitive states (Matheson & Frances, Reference Matheson, Frances and Zalta2018). Strongly embracing an egocentric bias might entail that one completely endorses their own opinion and consequently gives no weight to any peer’s opposing opinion.

On an alternative steadfast view, one might retain their initial opinion for the right reasons. The right reasons view appeals to all the evidence made available to each peer upon learning of a disagreement that includes first order evidence (the evidence available prior to learning of the disagreement that pertains to the disputed proposition(s)) and higher order evidence (evidence obtained upon learning of a disagreement that pertains to one’s reasoning on the matter). The higher order evidence one possesses is that they have a particular belief on the basis of the first order evidence and that their peer has a particular belief on the basis of that same first order evidence. By weighing the higher order evidence equally, peers are not commanded to conciliate or suspend judgement. Rather, weighed equally, the two facts cancel out, leaving each peer with exactly the same evidence before learning of their disagreement. Consequently, the peers retain their initial beliefs, and they do so for the right reasons (see Kelly, Reference Kelly, Hawthorne and Szabo2005).

However, Kelly (Reference Kelly, Goldman and Whitcomb2011) has since conceded that the balance between first order and higher order evidence is not always as straightforward as earlier and defends in more recent work the total evidence view. The question, though, is how should peers respond to a disagreement if the higher order evidence is not always swamped by the first order evidence? According to Kelly, there is no simple answer to the question. Should the learned higher order evidence speak strongly in favor of one having made a mistake in their reasoning, they ought to temper their confidence more than if the higher order evidence speaks less strongly in favor of one making a mistake. So, the answer is: it really depends on one’s total body of evidence, first order and higher order evidence (Kelly, Reference Kelly, Goldman and Whitcomb2011, 200–202). Proponents of the total evidence view thus reject conciliationism as the universally correct way of responding to a peer disagreement since on some occasions, one should hold steadfast in their answer, while on others, they ought to lose a whole lot of confidence. The equal weight view is mistaken then that there’s one and only one way that you should revise your opinion in response to a peer disagreement. The total evidence view can thus be seen as a middle ground between steadfastness and conciliationism.

Similarly, Jennifer Lackey’s (Reference Lackey, Haddock, Millar and Pritchard2010) justificationist view might be thought of along the same lines. On her view, what matters most, as the name suggests, is the justification of one’s belief. What sets the justificationist view apart from conciliationism is a denial of the independence condition, which Lackey shows to be counterintuitive through the following example.

As Lackey correctly suggests, you should not be rationally required to revise your opinion in the previous case. The overwhelming evidence you have supporting your belief and the personal information you possess about yourself provides evidence that Harry is mistaken. The disagreement, therefore, yields evidence, but not for your mathematical belief. Rather, the disagreement provides evidence for your belief about Harry, which conveys that, in fact, he’s not your peer (283). This case also seems to speak in favor of the correctness of the no independent weight view on occasions.

2.2 Group Rationality and Justified Group Belief

Notice that the problem of peer disagreement focuses on what individuals should come to believe. While we’re interested in how individuals ought to respond to peer disagreements, we’re equally interested in how individuals should form credences as a group. These are two different projects, but we contend that pooling strategies can help with both. We should note, however, that group credences generated by means of opinion pooling are often regarded as epistemic compromises, merely resulting from the group members’ opinions, but not necessarily consensus opinions.Footnote ⁹ That is, individuals come to agree on some set of opinions for group reasoning and decision-making purposes, but the members are not necessarily committed to holding those same opinions individually (see Wagner, Reference Wagner2010; Moss, Reference Moss2011).

Although epistemic compromises do not entail complete epistemic unity among a group’s members, we take them to sufficiently represent the attitudes of group agents, and this sentiment appears to be shared by others as we’ll come to see shortly. But what good are these group attitudes if they don’t guarantee complete epistemic solidarity among a group’s members? Following Dogramaci and Horowitz (Reference Dogramaci and Horowitz2016), we might say that conformity via compromise promotes (group) rationality, which gets groups closer to meeting their epistemic goal(s). This sentiment similarly resonates through Matthew Kopec’s (Reference Kopec2019) goal-oriented view.

We’ll follow Kopec on this goal-oriented view of group rationality, but we’ll broaden the scope of goals by including practical goals also.

In sticking with the epistemic dimension for now, what sort of epistemic goals might groups have? Dogramaci and Horowitz pointed out the obvious: getting to the truth.Footnote ¹⁰ But true belief, whether individually or collectively held, is not the only aim of rational believers. Ever since the days of Plato to the early modern philosophers, like Locke and Hume, epistemologists generally have come to accept that true belief must be supplemented in order to be rational. One supplement is justification. Thus, groups should aim to have justified true beliefs. Of course, achieving that is easier said than done. It’s hard enough for an individual to hit the epistemic target, let alone a group of individuals with different beliefs and possibly different evidence bases. That said, recent work on the justifiedness of group belief has become a focal point in social epistemology.

2.3 Group Rationality and Responsibility

Taken as agents in their own right, groups should not only aim to fulfill their epistemic goals, as Group Rationality requires, but their practical goals also. For why else should individuals be willing to compromise or reach a consensus if the group opinions of the group agent are not put to use in practice? If this is right, then Group Rationality should be broadened to cover the practical interests of groups also, which we’ll assume from here on.Footnote ¹⁵

Furthermore, Group Rationality leaves open ‘attitude’, which might generally mean intentional attitude. Intentional attitudes, for example, beliefs and desires, then ought to be the most instrumentally effective in meeting a group’s epistemic goals. But again, epistemic goals aren’t the only things groups care about and thus, intentional attitudes are collectively rational by being most instrumentally effective in also meeting a group’s practical goals. If correct, collective attitudes set up for group action as the pillars of means-end practical reasoning.Footnote ¹⁶ In turn, the actions performed by groups, like the actions performed by individuals, yield consequences that they can be held accountable for. Intentional group attitudes leading to group action make groups fit for responsibility.

3.1 The Formal Framework

Let’s begin with the individuals. To represent an individual’s opinions on a certain matter, we must represent the propositions about which they have an opinion as well as their opinions about them. Conforming to the standard parlance of philosophical decision theory and epistemology, we represent a proposition as a set of possible worlds. Until Section 7, we assume there is a finite set of possible worlds, $W=\{w_1,\dots ,w_n\}$, grained as finely as needed to represent the individual’s opinions, but no finer, and that the individual assigns credences to each element of the full algebra $A$ of subsets of $W$. Given $X$ and $Y$ in $A$, we write $X\vee Y$ for the union of $X$ and $Y$, $X\wedge Y$ for their intersection, and $\neg X$ for the complement of $X$.

Next, we represent the individuals’ opinions about these propositions by a function $P:A\to [0,1]$, which takes each proposition $X$ in $A$ and returns $P\left(X\right)$, the individual’s unconditional credence in $X$. Until Section 7, we assume $P$ satisfies the Kolmogorov axioms for probabilities: that is, $P\left(W\right)=1$, $P\left(\varnothing \right)=0$, and $P(X\vee Y)=P\left(X\right)+P\left(Y\right)$ whenever $X$ and $Y$ are disjoint.

We also write $P\left(X\mid Y\right)$ for the individual’s conditional credence in $X$ given $Y$, and define it using the so-called Ratio Formula: if $P\left(Y\right)>0$,

$P\left(X\mid Y\right)=\frac{P(X\wedge Y)}{P\left(Y\right)}$

Otherwise, it is undefined.

At various points, we will refer to Bayes’ Rule, which is the standard method for updating a probability function upon receipt of some new evidence that comes in the form of a proposition learned with certainty. It says that your new unconditional credence in a proposition should be your old conditional credence in it given the evidence you’ve learned. That is, for an individual with probability function $P$, upon learning $E$, their new credence in $X$ should be $P\left(X\mid E\right)$.

Turning our attention now to groups. Suppose we have a group of individuals $i=1,\dots ,n$. Until Section 7, we assume each individual $i$ has opinions that are represented by a probability function $P_i$ on the same agenda $A$. We write ${\Delta }_A$ for the set of all probability functions on $A$, and ${\Delta }_A^n$ for the set of sequences $(P_1,\dots ,P_n)$ of $n$ probability functions on $A$. We call such a sequence an opinion profile. A pooling strategy is a function $F:{\Delta }_A^n\to {\Delta }_A$. In other words, $F$ is a function that takes an opinion profile, which is a sequence of probability functions, one for each member of the group, to a single probability function, which represents the collective credences of the group. (When there is no ambiguity, we drop the subscript and write $\Delta$ and ${\Delta }^n$ instead of ${\Delta }_A$ and ${\Delta }_A^n$.)

In Section 4.4, we consider a more general representation of collective credences that uses imprecise probabilities.

With the basic framework now in place, we present a wish list of desirable criteria we would like a pooling strategy $F$ to meet before taking the plunge into the abyss of infinitely many pooling strategies. So, let us begin with some staples from the philosophical and statistical literatures that have been given as part of the so-called axiomatic approach.

3.2 Preserving Unanimous Judgments

The first concerns agreement. There are two versions; the first is strictly stronger than the second.

(Local) Unanimity Preservation. For all probability functions $P_1,\dots ,P_n$ on $A$, and for all propositions $X$ in $A$, if $P_1\left(X\right)=\cdots =P_n\left(X\right)$, then

$F(P_1,\dots ,P_n)\left(X\right)=P_1\left(X\right)=\cdots =P_n\left(X\right)$

That is, if all individuals agree about some proposition, the pool should agree with them about that proposition.

(Global) Unanimity Preservation. For all probability functions $P_1,\dots ,P_n$ on $A$, if, for all propositions $X\in A$, $P_1\left(X\right)=\cdots =P_n\left(X\right)$, then for all $X$ in $A$,

$F(P_1,\dots ,P_n)\left(X\right)=P_1\left(X\right)=\cdots =P_n\left(X\right)$

That is, if all individuals agree about all propositions, the pool should agree with them about those propositions.

If group members find themselves agreeing with one another, the unanimity criteria demand that the agreement is maintained after pooling. As we know, agreement is often hard to come by; these criteria say that groups should take advantage of it when they can. That being said, relaxing the constraints may be necessary if individuals have different private evidence (Dietrich & List, Reference Dietrich, List, Hájek and Hitchcock2015), since each might have evidence that supports a particular proposition strongly, but which, when combined, refutes it.

3.3 Eventwise Independence

The second criterion concerns the independence of a group’s credence in a proposition from the individual credences of all other propositions in the agenda:

Eventwise Independence.Footnote ²⁰ There exists a function $G:A\times [0,1{]}^n\to [0,1]$ such that for all probability functions $P_1,\dots ,P_n$ on $A$, and for all propositions $X$ in $A$,

$F(P_1,\dots ,P_n)\left(X\right)=G(X,P_1(X),\dots ,P_n(X\left)\right)$

This criterion is also known as the weak setwise function property, which is equivalent to the marginalization property (McConway, Reference McConway1981), and neutrality (Dietrich & List, Reference Dietrich and List2017). The idea is that the group credence for any proposition $X$ should depend only on the group members’ credences in $X$; it should not depend on their individual credences about anything else. It would be unusual, epistemically, if this weren’t the case. Say, for example, that the collective evaluates the propositions that a coin flip results in heads face up and that the global average surface temperature reaches a new record high in 2023. Whatever way the collective chooses to pool, the group credence for the coin landing heads should not depend on the members’ credences concerning the global average surface temperature.Footnote ²¹

3.4 Ruling Out Dictators

The third criterion is based on a principle of welfare economics (Arrow, Reference Arrow1951), but for our purposes, we’re interested in the epistemic implications:

Why care about Non-Dictatorship? Of course, it depends on why you are using the pooling function. If your purpose is to give a summary of the opinions among the individuals in the group, Non-Dictatorship lays down the plausible principle that no good summary depends only on a single individual’s view. Similarly, if you wish to use the pooled opinions to make a decision on behalf of the group that all members can get behind and for which they are happy to be held responsible, and if the members of the group share the same evidence, but have different posterior credences because they don’t share the same prior, then again Non-Dictatorship lays down a plausible principle. But there are also cases where we can legitimately violate it. Perhaps you will use the pooled opinion to choose on behalf of the group, and the group members all have the same prior, but one member has all the evidence that the others have and maybe even some more on top of that: perhaps they’re the head of an intelligence organization who sees all the intelligence reports, while all the other members of the group are intelligence officers who each see only some. Then in that case there seems no problem with electing them the dictator.

3.5 Bounding the Group’s Opinions

In the same vein, the fourth criterion might be seen as a constraint motivated by the same epistemic goal but more explicit in its direction:

Boundedness.Footnote ²² For all probability functions $P_1,\dots ,P_n$ on $A$, and for all propositions $X$ in $A$,

By treating group members’ credences as evidence, the group’s total evidence signals that the evidentially supported group credence lies somewhere in the range of individual credences. Boundedness ensures that pooling does not yield group credences outside of the range and thus respects the group’s total evidence. Besides respecting the evidence, empirical work on collective intelligence has demonstrated that groups often bracket the truth (Soll & Larrick, Reference Soll and Larrick2009). Thus, pooling strategies abiding by Boundedness promote Group Rationality in two ways: respecting the evidence and getting closer to the truth (more on the latter in Section 5). Furthermore, since Boundedness implies Unanimity Preservation, pooling rules satisfying the former consequently satisfy the latter.

While Boundedness appears to be an intuitive constraint, Easwaran et al. (Reference Easwaran, Fenton-Glynn, Hitchcock and Velasco2016) have appealed to an intuitive case by Christensen (Reference Christensen2009) to argue it isn’t always desirable. Suppose that a doctor is 97% confident about the right dosage of a drug to administer to a patient. The doctor learns that their equally qualified colleague is 96% confident about the same dosage for that patient. Christensen concludes that in this case, learning that a colleague is also very confident about the dosage should boost, not lower, the doctor’s confidence in the dosage being correct. If we take this intuition to hold in the context of pooling the two doctors’ opinions, the pooled opinion would indeed lie outside the range and to the right of the interval on the real line. And such violations of Boundedness are even more compelling when the individuals have different private information (Dietrich, Reference Dietrich2010).

3.6 The Interaction of Pooling and Learning

The next criterion concerns pooling and learning. We described earlier an application of Bayesian learning in assessing the reliability of others. Given its prominence in many scientific fields, and its dominance as a rule for learning in formal epistemology and decision and game theory, Bayes’ rule for updating on new information can be regarded as a fundamental norm for both individual and collective rationality in general – recall: it says that, upon learning a proposition $E$, your new unconditional credence in $X$ should be your old conditional credence in $X$ given $E$. What’s more, there are many arguments to the effect that it is the sole rational way for an individual or a group to update their credal opinions: updating in this way maximizes expected accuracy (Greaves & Wallace, Reference Greaves and Wallace2006); it is the only way to avoid being accuracy dominated (Briggs & Pettigrew, Reference Briggs and Pettigrew2020; Nielsen, Reference Nielsen2021); it is the only way to avoid being vulnerable to a diachronic Dutch Book (Lewis, Reference Lewis1999). In light of this, any pooling strategy adopted by a group should play nicely with Bayesian learning, especially if Group Rationality is taken to be part of the group’s ethos. One consequence of the interplay is that it should make no difference whether all individuals update on the same evidence and pool their updated credences, or whether they pool their individual credences and then update the pool on the evidence.

External Bayesianity. For all probability functions $P_1,\dots ,P_n$ on $A$ and for all propositions $X$ and $E$ in $A$,

$F\left(P_1\right(-\mid E),\dots ,P_n(-\mid E\left)\right)\left(X\right)=F(P_1,\dots ,P_n)\left(X\mid E\right)$

where $P_i(-\mid E)$ is the probability function obtained from $P_i$ by updating on $E$ using Bayes’ Rule.

That is, the following diagram should commute:

To drive the intuition, suppose you wish to determine whether a group is responsible for some harm caused by a decision that they made collectively – perhaps they’re the board of a corporation whose products caused injury and they’re liable if their corporate credence that this would happen is above some threshold. To do this, you need to determine what the group’s credences were in a range of relevant propositions, and you hope to ascribe those group credences, which attached to the collective agent that is the group, rather than to any of the individuals, by using a pooling strategy. Then it seems you’ll want that pooling function to satisfy External Bayesianity. If it doesn’t, it’s possible that it will ascribe different credences to the group depending on whether you update the individuals on the group’s evidence and then pool the posteriors or pool the priors and update them on the group’s evidence. And that sort of arbitrariness might well undermine your claim that the group is liable based on the credence you ascribe – perhaps they are if you use the group credences derived from pooling then update, but not if you update first then pool. And even if both give credences that render them liable, it’s plausible that the group is more liable, and therefore deserving of a more severe rebuke, the higher their credence. So the credence itself matters, not just whether it lies above some threshold.

3.7 Preserving Judgments of Independence

Our sixth criterion maintains that, after pooling, there should be no probabilistic correlation established between propositions that all members of the group judge beforehand to be probabilistically independent:

Probabilistic Independence Preservation. For all probability functions $P_1,\dots ,P_n$ on $A$, if, for all $i=1,\dots ,n$ and $X,Y$ in $A$ such that $P\left(Y\right)>0$, $P_i\left(X\mid Y\right)=P_i\left(X\right)$, then

$F(P_1,\dots ,P_n)\left(X\mid Y\right)=F(P_1,\dots ,P_n)\left(X\right)$

Probabilistic Independence Preservation, like Bayesian learning, shields groups from being Dutch Booked in some form. As Henry Kyburg and Michael Pittarelli (Reference Kyburg and Pittarelli1996) illustrated, if Probabilistic Independence Preservation is violated, then there exists a book of bets that the group will find acceptable but that guarantees they’ll lose money come what may.Footnote ²³ Whatever values each individual holds, it’s reasonable to assume that all group members prefer not to exchange bets on propositions in the agenda that result in such a loss. Indeed, any group that takes Group Rationality as part of its ethos should be committed to Probabilistic Independence Preservation, for its violation by a chosen pooling strategy strictly goes against it.

3.8 No Regrets

Similar in spirit, our final criterion concerns an expected and unnecessary loss from “throwing away money,” so to speak, by overpaying for bets or underselling them. Call the loss ex ante regret. Let each probability function $P_i$ for $i=1,\dots ,n$ induce a pair of regret functions, $r_{P_i}^{-}(\cdot ,\cdot )$ and $r_{P_i}^{+}(\cdot ,\cdot )$, that represent the ex ante regret of trading bets under $P_i$ that pay one monetary unit if $X\in A$ is true and zero otherwise. Each function maps a price-proposition pair $(u,X)\in [0,\infty )\times A$, relative to $P_i$, to a real number, and we assume throughout that they take the functional forms $r_{P_i}^{-}(u,X)=max\{u-P_i(X),0\}$ and $r_{P_i}^{+}(v,X)=max\left\{P_i\right(X)-v,0\}$.

For $u\in [0,\infty )$ and $X\in A$, $r_{P_i}^{-}(u,X)$ is the ex ante regret of buying a bet on $X$ for a price $u$ and $r_{P_i}^{+}(u,X)$ is the ex ante regret of selling a bet on $X$ for a price $u$, relative to $P_i$. Considering the ex ante regrets with respect to trading bets for all individuals $i=1,\dots ,n$, pooling strategies should ensure the following:

That’s to say all individuals $i$ should have no ex ante regrets of trading bets at the group’s maximum buying price $F(P_1,\dots ,P_n{)}^{-}(X)=minF(P_1,\dots ,P_n\left)\right(X)$ and minimum selling price $F(P_1,\dots ,P_n{)}^{+}(X)=maxF(P_1,\dots ,P_n\left)\right(X)$ for all $X\in A$. Given that the most common pooling strategies return a single probability function, $F(P_1,\dots ,P_n{)}^{-}(X)=F(P_1,\dots ,P_n{)}^{+}\left(X\right)=F(P_1,\dots ,P_n)\left(X\right)$.Footnote ²⁴ We’ll extend pooling strategies later on, where the pooling mechanism returns a set of probability functions that need not be a singleton set, that is, the credences may be imprecise.

From a practical standpoint, the No Regrets criterion should be quite intuitive, as no rational individual, in general, should be willing to pay a price above an asset’s perceived value when buying it, nor should the individual be willing to accept a price below the perceived value of the asset when selling it. In other words, individuals should neither happily expect to overpay for an asset nor undersell it. That just seems to be a bit of common sense. Although the constraint is quite strong, as it quantifies over all individuals, it is plausible when the individuals are epistemic peers (Elkin & Wheeler, Reference Elkin and Wheeler2018) or if the group is held liable for decisions made under the group’s credences since the group must answer for any bad deals made upon violating it when at least one individual signaled so much all along.

In the next section, we’ll attend to the pooling axioms provided here and learn, unsurprisingly, that not all of them play nicely with one another. But evaluating pooling strategies beyond mere impossibility results will be a goal of ours in the coming pages, illuminating the context-dependency of some criteria. This will be, in part, motivated by the earlier philosophical discussions. With any luck, our unifying approach might inspire others to look beyond mere technical difficulties and consider the instrumental value of pooling strategies based on the epistemic and practical interests and needs of groups in different circumstances.

4.1 Linear Pooling

Earlier, we met epidemiologists Anya, Bon, and Carys who are, respectively, 20%, 60%, and 70% sure that polio will be completely eradicated by 2030. How sure are they as a group? If they were to act as a group, perhaps by deciding how much funding a government should plan to spend on care for polio patients in the future, which credences should they use? If they were to be held responsible as a group, perhaps for that funding planning decision, which credences should we attribute to them? If we’re to summarize their views as a single credence to give some third party some information about views within the group, which should we use?

A natural answer to all of these questions is to say that we should take the average of their credences; or, more precisely, their arithmetic mean. That is, we should say that, as a group, Anya, Bon, and Carys have credence $\frac{1}{3}0.2+\frac{1}{3}0.6+\frac{1}{3}0.7=0.5$ that polio will be eradicated by 2030.

Here is the pooling strategy in greater generality:

Straight Linear Pooling Suppose $P_1,\dots ,P_n$ are probabilistic credence functions defined on the same agenda $A$. Then, for any $X$ in $A$,

$LP(P_1,\dots ,P_n)\left(X\right)=\frac{1}{n}{P}_1\left(X\right)+\cdots +\frac{1}{n}{P}_n\left(X\right)$

That is, the straight linear pool’s credence in $X$ is simply the average of the individuals’ credence in $X$. Notice that Straight Linear Pooling serves as a natural candidate for pooling under the equal weight view and may be epistemically motivated by such an account (Jehle & Fitelson, Reference Jehle and Fitelson2009).

But straight linear pooling is just one member of a larger family of strategies. In straight linear pooling, the same weight is afforded to each individual’s opinion. In the larger family, these weights can vary, which might be motivated by permissivist views like the justificationist and total evidence views. For instance, in the case of epidemiologists earlier, we might decide to give less weight to Carys, because her work is not directly on polio itself, while Anya’s and Bon’s is. So perhaps Anya receives weight $\frac{2}{5}$, rather than $\frac{1}{3}$, and Bon receives the same, but Carys receives $\frac{1}{5}$, rather than $\frac{1}{3}$. Then the weighted linear pool of their credences is $\frac{2}{5}0.2+\frac{2}{5}0.6+\frac{1}{5}0.7=0.46$.

So, to specify a specific pooling strategy in this family, we have to specify the weight that each individual will receive. A weight is a nonnegative real number, and the weights for all the individuals must add up to 1. So, if $P_1,\dots ,P_n$ are the probabilistic credence functions of the individuals in the group, then a sequence of weights is a sequence $\Lambda =({\lambda }_1,\dots ,{\lambda }_n)$, where $0\le {\lambda }_1,\dots ,{\lambda }_n\le 1$ and ${\lambda }_1+\cdots +{\lambda }_n=1$. In the previous paragraph, the sequence of weights for Anya, Bon, and Carys was $(\frac{2}{5},\frac{2}{5},\frac{1}{5})$. So here is the generalization of linear pooling, tracing back at least to Stone (Reference Stone1961):

Linear Pooling Suppose $P_1,\dots ,P_n$ are probabilistic credence functions defined on the same agenda $A$; and suppose $\Lambda =({\lambda }_1,\dots ,{\lambda }_n)$ is a sequence of weights. Then, for any $X$ in $A$,

${LP}^{\Lambda }(P_1,\dots ,P_n)\left(X\right)={\lambda }_1{P}_1\left(X\right)+\cdots +{\lambda }_n{P}_n\left(X\right)$

Note: If $P_1,\dots ,P_n$ are probabilistic credence functions, then ${LP}^{\Lambda }(P_1,\dots ,P_n)$ is a probabilistic credence function as well.

We say that ${LP}^{\Lambda }$ is dictatorial if one individual gets all the weight, that is, if ${\lambda }_i=1$ for some $i$ and ${\lambda }_j=0$ for all $j\ne i$. In that case, for any $X$ in $A$,

${LP}^{\Lambda }(P_1,\dots ,P_n)\left(X\right)=P_i\left(X\right)$

Individual $i$ is then a dictator for that group – what they say goes! Assuming Non-Dictatorship, such a pooling strategy must be ruled out, but on the other hand, the case of Harry from Section 2.1 provides a compelling instance where we might relax the constraint and accept the latter strategy. But let’s set that issue aside for now.

To evaluate the linear pooling strategies, let’s focus our attention on the axioms from Section 3 that these pooling strategies satisfy, which they don’t, and whether there are any combinations of these axioms that these pooling strategies alone satisfy.

Each linear pooling strategy ${LP}^{\Lambda }$ satisfies the following axioms:

1. (1a) Local Unanimity Preservation

   Why? The arithmetic mean of $n$ copies of the same number is just that number.

2. (1b) Global Unanimity Preservation

   Why? This follows from Local Unanimity Preservation.

3. (2) Eventwise Independence

   Why? The pooled credence in a proposition is always just the weighted average of the individual credences in that proposition. In order to calculate the linear pool’s credence in a proposition, you don’t need to know the individuals’ credences in anything other than that proposition.

4. (3) Boundedness

   Why? The weighted average of a sequence of numbers is always at least as great as the smallest in the sequence and at most as great as the largest.

Rather unsurprisingly, the only weighted linear pooling strategies that satisfy the following are non-dictatorial ones:

1. (4) Non-Dictatorship

   Why? The clue is in the name!

The only weighted linear pooling strategies that satisfy External Bayesianity and Probabilistic Independence Preservation are dictatorial:

1. (5) External Bayesianity

Why? This follows from a theorem by Madansky (Reference Madansky1964), which says that the only way to ensure that a linear pooling strategy will commute with Bayesian updating on evidence regardless of the individual credence functions it is fed is to make it dictatorial, so that there is one individual whose opinions always give the group’s opinions, no matter what they are, or what the other individuals think. To see that linear pooling violates External Bayesianity, it’s sufficient to give Madansky’s theorem for the case of two individuals.Footnote ²⁵

Theorem 1 (Madansky, 1964). Suppose that

1. (a) $P,Q$ are probabilistic credence functions defined on $A$,

2. (b) $\Lambda =(\lambda ,1-\lambda )$ is a sequence of weights,

3. (c) $X$ and $E$ are propositions in $A$,

4. (d) $P\left(E\right),Q\left(E\right)>0$,

Then, if

${LP}^{\Lambda }(P,Q)\left(X|E\right)={LP}^{\Lambda }\left(P\right(-\mid E),Q(-\mid E\left)\right)\left(X\right)$

Then at least one of the following must be true:

1. (i) $\Lambda$ is dictatorial. That is, $\lambda =0$ or $\lambda =1$.

2. (ii) $P\left(X|E\right)=Q\left(X|E\right)$. That is, all individuals agree on how likely $X$ is given $E$.

3. (iii) $P\left(E\right)=Q\left(E\right)$. That is, all individuals agree on how likely $E$ is.

Given that it is easy to specify $P,Q$, along with $X$ and $E$, such that (ii) and (iii) don’t hold, it follows that, if linear pooling with weights $\Lambda$ commutes with conditionalization, then (i) must be true and $\Lambda$ is dictatorial.

1. (6) Probabilistic Independence Preservation

Why? This follows from a theorem that is in the background in Laddaga (Reference Laddaga1977) and Lehrer & Wagner (Reference Lehrer and Wagner1983). It says that it is extremely rare for linear pooling to preserve judgments of independence, and only dictatorial versions guarantee it. A probabilistic credence function $P$ takes two propositions $X$ and $Y$ to be independent iff $P(X\wedge Y)=P\left(X\right)P\left(Y\right)$.Footnote ²⁶

Lastly, no linear pooling function satisfies

1. (7) No Regrets

   Why? Suppose that $P_i\ne P_j$ for some individuals $i,j$. Let $F^{-}(P_1,\dots ,P_n)\left(X\right)$ be the group’s maximum buying price for a bet on $X$ that pays 1 monetary unit if $X$ is true and 0 otherwise, and $F^{+}(P_1,\dots ,P_n)\left(X\right)$ be the group’s minimum selling price for the same bet. Following de Finetti (Reference de Finetti1974), any probability function $Q$ yields fair prices for bets on all $X\in A$. That is, the maximum buying price for a bet on $X$ is $Q\left(X\right)$ and likewise, the minimum selling price for the same bet is $Q\left(X\right)$. In other words, $Q\left(X\right)$ is a two-sided price for a bet on and against $X$. Thus, ${LP}^{\Lambda -}(P_1,\dots ,P_n)\left(X\right)={LP}^{\Lambda +}(P_1,\dots ,P_n)\left(X\right)={LP}^{\Lambda }(P_1,\dots ,P_n)\left(X\right)$ for all $X\in A$. Suppose that $P_i\left(X\right)<{LP}^{\Lambda }(P_1,\dots ,P_n)\left(X\right)<P_j\left(X\right)$. Then, $r_{P_i}^{-}\left({LP}^{\Lambda }\right(P_1,\dots ,P_n\left)\right(X),X)>0$, thus violating No Regrets. (In this case, we also have $r_{P_j}^{+}\left({LP}^{\Lambda }\right(P_1,\dots ,P_n\left)\right(X),X)>0$.)

That completes the list of axioms. Let’s wrap up our discussion of them by noting one of the central results in the area, which says that Local Unanimity Preservation and Eventwise Independence together characterize the family of weighted linear pooling strategies (Aczél & Wagner, Reference Aczél and Wagner1980; McConway, Reference McConway1981). That is,

Theorem 3 (Aczél & Wagner 1980; McConway 1981). If $F$ is a pooling strategy and $F$ satisfies Local Unanimity Preservation and Eventwise Independence, then there is a sequence of weights $\Lambda =({\lambda }_1,\dots ,{\lambda }_n)$ such that, for any probabilistic credence functions $P_1,\dots ,P_n$,

$F(P_1,\dots ,P_n)={LP}^{\Lambda }(P_1,\dots ,P_n)$

Like all characterization results, this one, together with the list of axioms that linear pooling strategies never satisfy, furnishes us with a number of impossibility theorems as corollaries. For instance, there can be no pooling strategy that satisfies Unanimity Preservation, Eventwise Independence, Non-Dictatorship, and External Bayesianity. After all, any one that satisfies the first two axioms is a linear pooling strategy, and the only such strategies that satisfy the fourth axiom are dictatorial and so violate the third axiom.

How concerning is such an impossibility theorem? That depends on how plausible the axioms are. We’ll return to Probabilistic Independence Preservation and No Regrets in due course, but let us briefly say why External Bayesianity might not be so desirable.

Daniel and Fang are at a horse race. Each has an unconditional credence for whether Sugar will win the race today, an unconditional credence for whether she won yesterday’s race, and conditional credences for whether she will win today given her placing in yesterday’s race. Daniel thinks it’s very likely she won yesterday, while Fangs think it’s very unlikely. Each of them is equally expert on the matter in hand, and so, before they learn about yesterday’s race, they should both receive the same weight when we aggregate them to give the group credences. But now they each pick up a newspaper and learn that Sugar won yesterday’s race. They both update their credences accordingly. We now ask what weights they should receive when we aggregate their new updated credences to give the group’s new updated credences: should we expect them to remain the same as before the evidence came in? A natural answer is that we shouldn’t. We should expect Daniel to receive higher weight after the evidence comes in, since he had a much higher credence in the proposition that was learned as evidence – his track record, initially the same as Fang’s, is now superior to theirs. If that’s right, External Bayesianity is not a feature we should want our pooling strategies to have, for it demands that weights remain the same before and after updating. What’s more, the following theorem due to Raiffa (Reference Raiffa1968) shows that linear pooling actually agrees with our intuitive response to the case of Daniel and Fang:Footnote ²⁷

Theorem 4 (Raiffa 1968). Suppose that

1. (a) $P,Q$ are probabilistic credence functions defined on $A$,

2. (b) $\Lambda =(\lambda ,1-\lambda )$ and ${\Lambda }^{\prime }=({\lambda }^{\prime },1-{\lambda }^{\prime })$ are sequences of weights,

3. (c) $X$ and $E$ are propositions in $A$,

4. (d) $P\left(E\right),Q\left(E\right)>0$,

And suppose that

${LP}^{\Lambda }(P,Q)\left(X|E\right)={LP}^{{\Lambda }^{\prime }}\left(P\right(-|E),Q(-|E\left)\right)\left(X\right)$

Then

${\lambda }^{\prime }=\frac{\lambda P\left(E\right)}{\lambda P\left(E\right)+(1-\lambda )Q\left(E\right)}$

and

$1-{\lambda }^{\prime }=\frac{(1-\lambda )Q\left(E\right)}{\lambda P\left(E\right)+(1-\lambda )Q\left(E\right)}$

That is, if the group’s credence before the evidence arrives is a linear pool of the individuals’ priors and the group’s credence after the evidence is a linear pool of their posteriors, and if the latter is obtained from the former by conditioning on the evidence, then the weight assigned to an individual afterwards is proportional to the weight before and their credence in the evidence.

The upshot: perhaps the failure to satisfy External Bayesianity does not deal a devastating blow against linear pooling. Violating Probabilistic Independence Preservation and No Regrets, however, is less benign, which we’ll come to in Section 6.

4.2 Geometric Pooling

Our next collection of pooling strategies belong to the family of multiplicative pooling rules. Unlike the linear pooling strategy that was our concern in the previous section, geometric pooling is defined in two stages.Footnote ²⁸ First, we pick a partition of the logical space and pool the credences assigned to the elements of that partition; secondly, we use those pooled credences to define the pooled credences in more coarse-grained propositions.

Let’s use Anya, Bon, and Carys again. To specify the geometric pool of their credences, we must specify a partition. So let’s take the two-cell partition that contains (i) Eradicated, which says polio will be eradicated by 2030, and (ii) Not Eradicated, which says it won’t. So the epidemiologists’ credences are as follows:

As we saw in the previous section, to obtain the linear pool of their credences in Eradicated, we simply take their arithmetic mean, that is, $\frac{1}{3}0.2+\frac{1}{3}0.6+\frac{1}{3}0.7=0.5$, and to obtain the linear pool of their credences in Not Eradicated, we take their arithmetic mean as well, that is $\frac{1}{3}0.8+\frac{1}{3}0.4+\frac{1}{3}0.3=0.5$. And notice that the linear pool of a set of probabilistic credence functions is always guaranteed to be probabilistic itself.

To obtain the geometric pool of their credences in Eradicated, we start by taking their geometric mean, that is, ${0.2}^{\frac{1}{3}}\times {0.6}^{\frac{1}{3}}\times {0.7}^{\frac{1}{3}}\approx 0.438$. But if we did nothing else and also took the geometric pool of their credences in Not Eradicated to be the geometric mean of their credences in that proposition, that is, ${0.8}^{\frac{1}{3}}\times {0.4}^{\frac{1}{3}}\times {0.3}^{\frac{1}{3}}\approx 0.458$, we’d end up with non-probabilistic credences, since these two numbers don’t sum to 1. So instead of taking the geometric pool of some credences to be their geometric mean, we instead take it to be the normalized geometric mean. That is, we take the geometric means of the credences in each cell of the partition, and then multiply each of these geometric means by the same factor to ensure that the results sum to 1, as credences over a partition are required to do. The pooled credence in any proposition that is a disjunction of elements of the partition is then the sum of the pooled credences in the disjuncts.

Straight Geometric Pooling Suppose:

1. (i) $P_1,\dots ,P_n$ are probabilistic credence functions defined on $A$, and

2. (ii) $S\subseteq A$ is a partition.

Then, first: for each $S$ in $S$,

$\begin{array}{cc}{GP}_S(P_1,\dots ,P_n)\left(S\right)& =\frac{\sqrt[n]{P_1\left(S\right)\times \cdots \times P_n\left(S\right)}}{\sum _{S^{\prime }\in S}\sqrt[n]{P_1\left(S^{\prime }\right)\times \cdots \times P_n\left(S^{\prime }\right)}}\\ & =\frac{\prod _{i=1}^n{P}_i(S{)}^{\frac{1}{n}}}{\sum _{S^{\prime }\in S}\prod _{i=1}^n{P}_i(S^{\prime }{)}^{\frac{1}{n}}}\end{array}$

And, second: for any $X$ in $A$ that is a disjunction of propositions from $S$,

${GP}_S(P_1,\dots ,P_n)\left(X\right)=\sum _{\begin{array}{c}S\in S\\ S\subseteq X\end{array}}{GP}_S(P_1,\dots ,P_n)\left(S\right)$

One immediate consequence of this definition is that, if we are to pool the credences of a group of individuals who share the agenda $A$, then there must be a partition $S\subseteq A$ such that every proposition in $A$ is a disjunction of some propositions from this partition. We cannot pool the opinions of a group who have credences only in The die will land on six, The die will land on an even number, The die will land on a number less than 4. Of course, this is guaranteed if we assume, as we have done throughout, that $A$ is the full algebra of subsets of a finite set of possible worlds $W$.

Another consequence is that the geometric pool of some credences is sensitive to the partition you choose. Suppose Rodrigo and Stefan have the following credences:

	Heavy Rain	Light Rain	No Rain
Rodrigo ($P_R$)	$\frac{1}{6}$	$\frac{1}{3}$	$\frac{1}{2}$
Stefan ($P_S$)	$\frac{1}{3}$	$\frac{1}{6}$	$\frac{1}{2}$

Then start with the partition $S=\{\text{Heavy Rain},\text{Light Rain},\text{No Rain}\}$. Then

	Heavy Rain	Light Rain	No Rain
${GP}_S(P_R,P_S)$	0.24	0.24	0.52

Next, the partition $S^{\prime }=\{\text{Rain},\text{No Rain}\}$. Then

	Heavy Rain	Light Rain	No Rain
${GP}_{S^{\prime }}(P_R,P_S)$	0.25	0.25	0.5

In fact, just as there is a whole family of linear pooling strategies, each fixed by a sequence of weights, so there is a whole family of geometric pooling strategies:

Geometric Pooling Suppose:

1. (i) $P_1,\dots ,P_n$ are probabilistic credence functions defined on $A$,

2. (ii) $S\subseteq A$ is a partition, and

3. (iii) $\Lambda =({\lambda }_1,\dots ,{\lambda }_n)$ is a sequence of nonnegative real numbers that sum to 1.

Then, first: for each $S$ in $S$,

${GP}_S^{\Lambda }(P_1,\dots ,P_n)\left(S\right)=\frac{P_1(S{)}^{{\lambda }_1}\times \cdots \times P_n(S{)}^{{\lambda }_n}}{\sum _{S^{\prime }\in S}{P}_1(S^{\prime }{)}^{{\lambda }_1}\times \cdots \times P_n(S^{\prime }{)}^{{\lambda }_n}}$

$=\frac{\prod _{i=1}^n{P}_i(S{)}^{{\lambda }_i}}{\sum _{S^{\prime }\in S}\prod _{i=1}^n{P}_i(S^{\prime }{)}^{{\lambda }_i}}$

And, second: for any $X$ in $A$ that is a disjunction of propositions from $S$,

${GP}_S^{\Lambda }(P_1,\dots ,P_n)\left(X\right)=\sum _{\begin{array}{c}S\in S\\ S\subseteq X\end{array}}{GP}_S^{\Lambda }(P_1,\dots ,P_n)\left(S\right)$

4.3 Multiplicative Pooling

In fact, even the family of geometric pooling strategies is just one in a larger group of families known as the weighted multiplicative pooling strategies. Where geometric pooling uses sequences of weights that sum to 1, multiplicative pooling strategies use sequences that sum to any positive real number, though the most common is $n$, where $n$ is the number of individuals pooled. So we have:

Multiplicative Pooling Suppose:

1. (i) $P_1,\dots ,P_n$ are probabilistic credence functions defined on $A$,

2. (ii) $S\subseteq A$ is a partition, and

3. (iii) $\Lambda =({\lambda }_1,\dots ,{\lambda }_n)$ is a sequence of nonnegative real numbers.

Then, first: for each $S$ in $S$,

$\begin{array}{cc}{MP}_S^{\Lambda }(P_1,\dots ,P_n)\left(S\right)& =\frac{P_1(S{)}^{{\lambda }_1}\times \cdots \times P_n(S{)}^{{\lambda }_n}}{\sum _{S^{\prime }\in S}{P}_1(S^{\prime }{)}^{{\lambda }_1}\times \cdots \times P_n(S^{\prime }{)}^{{\lambda }_n}}\\ & =\frac{\prod _{i=1}^n{P}_i(S{)}^{{\lambda }_i}}{\sum _{S^{\prime }\in S}\prod _{i=1}^n{P}_i(S^{\prime }{)}^{{\lambda }_i}}\end{array}$

And, for any $X$ in $A$ that is a disjunction of propositions from $S$,

${MP}_S^{\Lambda }(P_1,\dots ,P_n)\left(X\right)=\sum _{\begin{array}{c}S\in S\\ S\subseteq X\end{array}}{GP}_S(P_1,\dots ,P_n)\left(S\right)$

As we did with linear pooling, let’s see which of our axioms geometric and multiplicative pooling strategies satisfy.

1. (1a) Local Unanimity Preservation

   No multiplicative pooling strategy satisfies this. Indeed, we saw an example earlier, in which Rodrigo and Stefan both gave credence $0.5$ to No Rain, but when we pooled over the more fine-grained partition of Light Rain, Heavy Rain, and No Rain, their pooled credence for No Rain was $0.52$.

2. (1b) Global Unanimity Preservation

   Any weighted geometric pooling strategy satisfies this, but no other multiplicative strategy. The problem is that

   ${MP}_S^{\Lambda }(P,\dots ,P)\left(S\right)=\frac{P(S{)}^{{\lambda }_1+\cdots +{\lambda }_n}}{\sum _{S^{\prime }\in S}P(S^{\prime }{)}^{{\lambda }_1+\cdots +{\lambda }_n}}$
   And, if ${\lambda }_1+\cdots +{\lambda }_n\ne 1$, it is easy to find $P$ such that this is not equal to $P\left(S\right)$ for all $S$ in $S$.

3. (2) Eventwise Independence

   None of the multiplicative pooling strategies satisfy this. While the numerator in the definition of ${MP}_S^{\Lambda }(P_1,\dots ,P_n)\left(S\right)$ depends only on the individuals’ credences $P_1\left(S\right),\dots ,P_n\left(S\right)$ in $S$, the denominator that normalizes this credence and ensures that the credences in the elements of the partition sum to 1 depends on the individuals’ credences in the other elements of the partition.

4. (3) Boundedness

   Any pooling strategy that violates Local Unanimity Preservation will violate Boundedness as well. And, in particular, Rodrigo and Stefan’s pooled credence in No Rain provides an example again. The pooled credence of $0.52$ does not lie between the maximum individual credence of $0.5$ and the minimum individual credence of $0.5$.

5. (4) Non-Dictatorship

   As in the case of linear pooling, any multiplicative rule for which ${\lambda }_i,{\lambda }_j>0$ for $i\ne j$ is non-dictatorial.

6. (5) External Bayesianity

   Perhaps the most notable feature of the multiplicative rules is that they do satisfy External Bayesianity.

7. (6) Probabilistic Independence Preservation

   No non-dictatorial multiplicative rule satisfies this.

8. (7) No Regrets

   Like linear pooling, no multiplicative rule satisfies No Regrets. This is made obvious by the multiplicative rules violating Boundedness. Since ${GP}_S^{\Lambda }(P_1,\dots ,P_n)\left(X\right)$ and ${MP}_S^{\Lambda }(P_1,\dots ,P_n)\left(X\right)$ can lie outside of $[min(P_1\left(X\right),\dots ,P_n\left(X\right)),max(P_1\left(X\right),\dots ,P_n\left(X\right)\left)\right]$ for some $X\in A$, either $r_{P_i}^{-}\left({GP}_S^{\Lambda }\right(P_1,\dots ,P_n\left)\right(X),X)>0$ or $r_{P_i}^{+}\left({GP}_S^{\Lambda }\right(P_1,\dots ,P_n\left)\right(X),X)>0$ and $r_{P_i}^{-}\left({MP}_S^{\Lambda }\right(P_1,\dots ,P_n\left)\right(X),X)>0$ or $r_{P_i}^{+}\left({MP}_S^{\Lambda }\right(P_1,\dots ,P_n\left)\right(X),X)>0$ for all $i$. If the group credences lie in $[min(P_1\left(X\right),\dots ,P_n\left(X\right)),max(P_1\left(X\right),\dots$, $P_n\left(X\right)\left)\right]$ for all $X\in A$, then the same holds for ${GP}_S^{\Lambda }$ and ${MP}_S^{\Lambda }$ as ${LP}^{\Lambda }$.

Let’s quickly take stock before moving on to imprecise group credences. What has been observed so far is that the linear and multiplicative pooling strategies fail to meet all of the desirable criteria we’d like them to. Take either the entire set of criteria or particular subsets, and we end up with some impossibility results. Notably, the linear and multiplicative rules fare poorly with the pragmatically oriented axioms, Probabilistic Independence Preservation and No Regrets. Given the discussion on group responsibility in Section 2.3, failure to satisfy the latter criteria might hint that it’s impossible to provide the sorts of group credences required for an adequate account of collective responsibility. In Section 6, we’ll better motivate the axioms and revisit this point.

4.4 Imprecise Probability Pooling

Up until now, we’ve considered “classical” pooling strategies that map opinion profiles to a single probability function. Rather than mapping sets of credences in propositions to precise, point-valued probabilities, we could pool credences using one of the many tools of imprecise probability (IP), giving us sets of probabilities instead, for example, $[0.2,0.7]$. On such occasions, we’ll use pooling functions taking the form of $F:{\Delta }_A^n\to P\left({\Delta }_A\right)$, where $P\left({\Delta }_A\right)$ is the power set of ${\Delta }_A$, which yield sets of probability functions representing the credences of groups.Footnote ²⁹

A group’s credences under this account can simply be summarized by a collective lower probability

$F^{-}(P_1,\dots ,P_n)\left(X\right)=inf\left\{P\right(X):P\in F\}\forall X\in A,$

which induces a collective upper probability,

$F^{+}(P_1,\dots ,P_n)\left(X\right)=sup\left\{P\right(X):P\in F\}\forall X\in A,$

through a conjugacy relation: $F^{+}(P_1,\dots ,P_n)\left(X\right)=1-F^{-}(P_1,\dots ,P_n)\left(\neg X\right)$.Footnote ³⁰ While $F^{-}$ and $F^{+}$ may sufficiently summarize a group’s credences, they alone don’t tell us the whole story. That’s why we suggest that IP pooling models also include the set $F(P_1,\dots ,P_n)$.Footnote ³¹ Throughout, we’ll focus on the properties of $F$. However, that doesn’t mean that collective lower probability isn’t useful. We’ll show that it indeed has an important role. Following the likes of Peter Walley (Reference Walley1991), we give $F^{-}$ a behavioral interpretation. Given a bet on $X$ that pays $1$ monetary unit if $X$ is true and $0$ otherwise for all $X\in A$, $F^{-}(P_1,\dots ,P_n)\left(X\right)$ is said to be a group’s supremum acceptable buying price for the bet, and $F^{+}(P_1,\dots ,P_n)\left(X\right)$ the group’s infimum acceptable selling price. Thus, the pair $F^{-}$ and $F^{+}$ have significance under the behavioral interpretation and are central components in the elicitation of group credences in uncertain propositions.

With one type of imprecise probability framework at hand that we might exploit in forming group credences,Footnote ³² here’s a natural candidate for pooling:

Convex Imprecise Probability Pooling   Suppose that $P_1,\dots ,P_n$ are probabilistic credence functions defined on the same agenda $A$.

$\text{C}(P_1,\dots ,P_n)=\text{conv}\{P_i:i=1,\dots ,n\},$

where ‘conv’ is the convex hull operator. Accordingly, C returns the convex hull of credences for all propositions $X\in A$. (The range of the function is consequently restricted to $D\subset P\left({\Delta }_A\right)$, which is the set of all convex sets of probability functions.) The given aggregation rule was considered in the past by Isaac Levi (Reference Levi1985) in forming a consensus between Bayes agents and has been more recently studied systematically by Rush Stewart and Ignacio Ojea Quintana (Reference Stewart and Quintana2018), whose approach we’ll often follow.Footnote ³³

Some, however, might be curious about the convexity property of the pooling strategy. Convexity, so it seems, is as contentious as how to pool, at least as far as the literature on imprecise probability is concerned. Some see it as a mathematical convenience (e.g., Walley, Reference Walley1991), while others find it philosophically motivated (e.g., Levi, Reference Levi1974, Reference Levi1985; Stewart & Quintana, Reference Stewart and Quintana2018). Levi (Reference Levi1985), for example, appears somewhat sympathetic to the linear pooling rule as a way of forming a compromise. However, he insists that in case of a disagreement, there is no uniquely rational (linear) compromise that individuals or groups are warranted in adopting. Rather, the set of all possible linear pools must be considered credible, but due to conflict, they are held in suspense. In other words, individuals and groups should suspend judgment on all possible credal (linear) compromises. And this commits individuals and groups to adopting the convex hull in their credal judgments provided that all credal distributions are permissible for calculating expectations prior to resolving the conflict through further inquiry.

Of course, not everyone is compelled by Levi’s view, but setting aside the philosophical motivations and mathematical considerations, convexity seems quite natural. However, there are practical reasons for why convexity should be abandoned that we’ll come to in Section 6. These reasons will also show us why we can’t rely solely on lower and upper probabilities. Alternatively, we’ll introduce a pooling strategy shortly that relaxes the convexity constraint and that foreshadows the pragmatic concerns convexity gives rise to. But first, let’s attend to the given IP pooling strategy and evaluate it under extended versions of our axioms.Footnote ³⁴

1. (1) [(1a)] Local Unanimity Preservation

   If all group members $i=1,\dots ,n$ have a shared probability function, $P$, then $\text{C}(P_1,\dots ,P_n)\left(X\right)=\left\{P\right(X\left)\right\}$ for all $X\in A$. Why? Because $\text{conv}\left\{P\right\}=\left\{P\right\}$.

2. (1b) Global Unanimity Preservation

   This follows from (1a).

3. (2) Eventwise Independence

Here’s a sketch. Suppose there’s a function $G:A\times [0,1{]}^n\to P([0,1])$. Let $G(X,P_1(X),\dots ,P_n(X\left)\right)=\text{conv}\left\{P_i\right(X):i=1,\dots ,n\}$ for all $X\in A$. For any $P\left(X\right)\in \text{conv}\left\{P_i\right(X):i=1,\dots ,n\}$, $P\left(X\right)$ is some convex combination of $P_i\left(X\right)$ for $i=1,\dots ,n$. Given the definition of C, $\text{C}(P_1,\dots ,P_n)\left(X\right)=\text{conv}\left\{P_i\right(X):i=1,\dots ,n\}$ for all $X\in A$. So, any $P\left(X\right)\in \text{C}(P_1,\dots ,P_n)\left(X\right)$ is some convex combination of $P_i\left(X\right)$ for $i=1,\dots ,n$. It follows that $\text{C}(P_1,\dots ,P_n)\left(X\right)=G(X,P_1(X),\dots ,P_n(X\left)\right)$. $\text{C}(P_1,\dots ,P_n)\left(X\right)$ thus only depends on the $P_i\left(X\right)$’s for all $X\in A$ and probability functions $P_1,\dots ,P_n$.Footnote ³⁵

1. (3) Boundedness

The IP pooling strategy trivially satisfies Boundedness. Since for all $X\in A$ and probability functions $P_1,\dots ,P_n$, $inf\text{C}(P_1,\dots ,P_n)\left(X\right)=min\left(P_1\right(X),\dots ,P_n(X\left)\right)$ and $sup\text{C}(P_1,\dots ,P_n)\left(X\right)=max\left(P_1\right(X),\dots$, $P_n\left(X\right))$, $\text{C}(P_1,\dots ,P_n)\left(X\right)\subseteq [min(P_1\left(X\right),\dots ,P_n\left(X\right)),max(P_1\left(X\right),\dots$, $P_n\left(X\right)\left)\right]$.Footnote ³⁶